Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1136 Structured version   Unicode version

Theorem bnj1136 29795
Description: Technical lemma for bnj69 29808. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1136.1  |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
bnj1136.2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1136.3  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
Assertion
Ref Expression
bnj1136  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
Distinct variable groups:    y, A    y, R    y, X
Allowed substitution hints:    th( y)    ta( y)    B( y)

Proof of Theorem bnj1136
StepHypRef Expression
1 bnj1136.2 . . . 4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
21biimpri 209 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  th )
3 bnj1136.1 . . . . 5  |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )
4 bnj1148 29794 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  e.  _V )
5 bnj893 29728 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  e.  _V )
6 simp1 1005 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  R  FrSe  A )
7 bnj1127 29789 . . . . . . . . . . 11  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  y  e.  A )
873ad2ant3 1028 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  y  e.  A )
9 bnj893 29728 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  y  e.  A )  ->  trCl ( y ,  A ,  R )  e.  _V )
106, 8, 9syl2anc 665 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  e.  _V )
11103expia 1207 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  e.  _V ) )
1211ralrimiv 2835 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  e.  _V )
13 iunexg 6775 . . . . . . 7  |-  ( ( 
trCl ( X ,  A ,  R )  e.  _V  /\  A. y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R )  e.  _V )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  e.  _V )
145, 12, 13syl2anc 665 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  e.  _V )
154, 14bnj1149 29593 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  e. 
_V )
163, 15syl5eqel 2512 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  e.  _V )
173bnj1137 29793 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
)
183bnj931 29571 . . . . 5  |-  pred ( X ,  A ,  R )  C_  B
1918a1i 11 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_  B )
20 bnj1136.3 . . . 4  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
2116, 17, 19, 20syl3anbrc 1189 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ta )
221, 20bnj1124 29786 . . 3  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
232, 21, 22syl2anc 665 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  C_  B )
24 bnj906 29730 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  pred ( X ,  A ,  R )  C_ 
trCl ( X ,  A ,  R )
)
25 bnj1125 29790 . . . . . . 7  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
) )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
26253expia 1207 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( y  e.  trCl ( X ,  A ,  R )  ->  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) ) )
2726ralrimiv 2835 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
28 ss2iun 4309 . . . . . 6  |-  ( A. y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( X ,  A ,  R ) )
29 bnj1143 29591 . . . . . 6  |-  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( X ,  A ,  R )  C_  trCl ( X ,  A ,  R )
3028, 29syl6ss 3473 . . . . 5  |-  ( A. y  e.  trCl  ( X ,  A ,  R
)  trCl ( y ,  A ,  R ) 
C_  trCl ( X ,  A ,  R )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
3127, 30syl 17 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
3224, 31unssd 3639 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl (
y ,  A ,  R ) )  C_  trCl ( X ,  A ,  R ) )
333, 32syl5eqss 3505 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  C_  trCl ( X ,  A ,  R
) )
3423, 33eqssd 3478 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078    u. cun 3431    C_ wss 3433   U_ciun 4293    predc-bnj14 29482    FrSe w-bnj15 29486    trClc-bnj18 29488    TrFow-bnj19 29490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-reg 8105  ax-inf2 8144
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-om 6699  df-1o 7182  df-bnj17 29481  df-bnj14 29483  df-bnj13 29485  df-bnj15 29487  df-bnj18 29489  df-bnj19 29491
This theorem is referenced by:  bnj1408  29834
  Copyright terms: Public domain W3C validator